Question Number 86671 by M±th+et£s last updated on 30/Mar/20 $$\int{sin}\left({x}\right)\:{arcsin}\left({x}\right) \\ $$ Answered by Rio Michael last updated on 30/Mar/20 $$\mathrm{let}\:\mathrm{me}\:\mathrm{give}\:\mathrm{a}\:\mathrm{try}. \\ $$$$\:\int\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\left({x}\right)\:{dx} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{taylor}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\:\left({x}\right)\:\mathrm{centred}\:\mathrm{at}\:\mathrm{0}…
Question Number 152203 by peter frank last updated on 26/Aug/21 $$\mathrm{If}\:\mathrm{x}\:\mathrm{is}\:\mathrm{real}\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{2}+\mathrm{i}\right)^{\left(\mathrm{1}+\mathrm{3i}\right)\mathrm{x}} +\left(\mathrm{2}−\mathrm{i}\right)^{\left(\mathrm{1}−\mathrm{3i}\right)\mathrm{x}} \\ $$$$\mathrm{is}\:\mathrm{also}\:\mathrm{real} \\ $$ Commented by MJS_new last updated on 26/Aug/21…
Question Number 152201 by mnjuly1970 last updated on 26/Aug/21 $$ \\ $$$$\:\:\:…\mathrm{Integral}… \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).{tan}^{\:−\mathrm{1}} \left({cot}\left({x}\right)\right){dx}\overset{?} {=}\:\mathrm{0} \\ $$$$\:\:\:\:\:{proof}\:::\:…. \\ $$$$\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).\:{tan}^{\:−\mathrm{1}} \left(\:{tan}\left(\frac{\pi}{\mathrm{2}}\:−{x}\:\right)\right){dx}…
Question Number 152186 by Tawa11 last updated on 26/Aug/21 $$\int\mathrm{x}^{\mathrm{n}} \:\mathrm{cos}\left(\mathrm{nx}\right)\:\mathrm{dx} \\ $$ Answered by mindispower last updated on 26/Aug/21 $${nx}={y} \\ $$$$\Leftrightarrow\frac{\mathrm{1}}{{n}^{{n}+\mathrm{1}} }\int{y}^{{n}} {cos}\left({x}\right){dx}…
Question Number 86646 by Ar Brandon last updated on 30/Mar/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$ Commented by mathmax…
Question Number 86643 by mathmax by abdo last updated on 29/Mar/20 $${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{ch}\left({x}\right)\right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 86638 by M±th+et£s last updated on 29/Mar/20 $${I}=\int\frac{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}{x}−\mathrm{1}}{\left({x}−\mathrm{3}\right)^{\mathrm{3}} \left({x}−\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$ Answered by MJS last updated on 29/Mar/20 $$\frac{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}}…
Question Number 152165 by peter frank last updated on 26/Aug/21 $$\int\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{3x}}\mathrm{dx} \\ $$ Answered by john_santu last updated on 26/Aug/21 $$\:\mathrm{I}=\int\:\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{3sin}\:\mathrm{x}−\mathrm{4sin}\:^{\mathrm{3}} \mathrm{x}}\:\mathrm{dx}\: \\ $$$$\mathrm{I}=\int\:\frac{\mathrm{dx}}{\mathrm{3}−\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}}…
Question Number 152161 by peter frank last updated on 26/Aug/21 $$\int\mathrm{cos}\:\left(\mathrm{log}\:\mathrm{x}\right)\mathrm{dx} \\ $$ Answered by Olaf_Thorendsen last updated on 26/Aug/21 $$\mathrm{F}\left({x}\right)\:=\:\int\mathrm{cos}\left(\mathrm{log}{x}\right)\:{dx} \\ $$$$\mathrm{F}\left({e}^{{u}} \right)\:=\:\int\mathrm{cos}{u}\:{e}^{{u}} {du}…
Question Number 152164 by peter frank last updated on 26/Aug/21 $$\int\frac{\mathrm{2x}+\mathrm{1}}{\mathrm{4}−\mathrm{3x}−\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$ Commented by Paradoxical last updated on 26/Aug/21 Commented by peter frank…